Comparing and Ordering Rational Numbers | Grade 8 Math | LevelKite

A rational number is any number you can write as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is not zero. The denominator cannot be zero because division by zero is undefined. Fractions such as $\frac{3}{4}$ and $-\frac{5}{6}$ are rational. So are decimals such as $-1.2 = \frac{-12}{10}$ and integers such as $7 = \frac{7}{1}$ and $0 = \frac{0}{1}$ .

In this lesson you will compare rational numbers. That means deciding which of two numbers is less, which is greater, or whether they are equal. For example, you will decide which of $-\frac{3}{4}$ and $-\frac{5}{6}$ is greater. You will also order rational numbers, which means arranging a list from least to greatest or from greatest to least.

Order comes from position. Think of a number line with $0$ in the middle, positive numbers to the right of $0$ and negative numbers to the left of $0$ . Every rational number has exactly one place on this line. For any two rational numbers $x$ and $y$ :

$x < y$ means $x$ is to the left of $y$ .
$x > y$ means $x$ is to the right of $y$ .
$x = y$ means $x$ and $y$ are at the same place.

This gives you a quick first check. Every negative number is to the left of $0$ and every positive number is to the right of $0$ . So any negative number is less than $0$ , and $0$ is less than any positive number. For example, $-\frac{9}{2} < \frac{1}{100}$ , and you can tell without any arithmetic.

When two numbers have the same sign, the signs cannot decide. Then you write both numbers in the same form and compare.

One useful form is fractions with the same positive denominator. The denominator gives the size of one step: a denominator of $12$ means steps of size $\frac{1}{12}$ . The numerator counts the steps from $0$ , to the right when it is positive and to the left when it is negative. So $\frac{7}{12}$ is $7$ steps to the right of $0$ . And $\frac{-7}{12}$ is $7$ steps to the left of $0$ , which is the place of $-\frac{7}{12}$ . That is why you can write a negative fraction with its sign in the numerator and keep the denominator positive.

When two fractions have the same positive denominator, their steps are the same size, so the numerators decide. The greater numerator lands farther right, so its fraction is greater: $\frac{5}{12} < \frac{7}{12}$ because $5 < 7$ , and $\frac{-7}{12} < \frac{-5}{12}$ because $-7 < -5$ . Equal numerators land on the same place, so the fractions are equal.

To give two fractions the same denominator, multiply the numerator and the denominator of a fraction by the same nonzero number. That is the same as multiplying the fraction by $1$ , because a nonzero number divided by itself is $1$ . So the value stays the same: $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ . Use a positive multiplier so that the denominator stays positive. Aim for a common denominator, which is a positive number that is a multiple of both denominators.

Worked example

Comparing two negative fractions

Which number is greater, $-\frac{3}{4}$ or $-\frac{5}{6}$ ?

Check the signs first. Both numbers are negative, so the signs alone cannot decide. Write each number with its negative sign in the numerator. Each number keeps its place on the number line, and both denominators are positive.
$-\frac{3}{4} = \frac{-3}{4}, \qquad -\frac{5}{6} = \frac{-5}{6}$
Choose a common denominator so that both fractions use steps of the same size. The least positive number that is a multiple of both 4 and 6 is 12.
$4 \times 3 = 12, \qquad 6 \times 2 = 12$
Rewrite each fraction with denominator 12. Multiply its numerator and its denominator by the same positive number. That is multiplying by 1, so each value stays the same.
$\frac{-3}{4} = \frac{-3 \times 3}{4 \times 3} = \frac{-9}{12}, \qquad \frac{-5}{6} = \frac{-5 \times 2}{6 \times 2} = \frac{-10}{12}$
The denominators are now the same positive number, so the numerators decide. Compare them as integers: negative 10 is to the left of negative 9, so negative 10 is less.
$-10 < -9, \quad \text{so} \quad \frac{-10}{12} < \frac{-9}{12}$
Write each fraction in its original form again. The order stays the same because each pair of forms names the same number.
$-\frac{5}{6} < -\frac{3}{4}$

Answer

$-\frac{3}{4}$ is greater than $-\frac{5}{6}$ , because $\frac{-9}{12}$ is to the right of $\frac{-10}{12}$ on the number line.

The other useful form is decimals. It helps most when a list mixes fractions and decimals. To write a fraction as a decimal, divide the numerator by the denominator, which is never zero. For a rational number, one of two things happens:

The division ends. You get a terminating decimal, such as $\frac{3}{4} = 0.75$ .
The division never ends, but a block of digits repeats forever. You get a repeating decimal, such as $\frac{2}{3} = 0.666\ldots$ , where the digit $6$ repeats. Put a bar over the block that repeats: $\frac{2}{3} = 0.\overline{6}$ .

For a negative fraction, find the decimal for the matching positive fraction and then write the negative sign: $\frac{5}{4} = 1.25$ , so $-\frac{5}{4} = -1.25$ .

Exact decimal forms and values to the nearest hundredth

Number	Exact decimal form	Type of decimal	To the nearest hundredth
$\frac{3}{4}$	$0.75$	Terminating	$\frac{3}{4} = 0.75$
$\frac{3}{5}$	$0.6$	Terminating	$\frac{3}{5} = 0.60$
$-\frac{5}{4}$	$-1.25$	Terminating	$-\frac{5}{4} = -1.25$
$\frac{2}{3}$	$0.\overline{6}$	Repeating	$\frac{2}{3} \approx 0.67$
$-\frac{4}{3}$	$-1.\overline{3}$	Repeating	$-\frac{4}{3} \approx -1.33$

To compare two positive decimals, line up the decimal points so that digits with the same place value are in the same column. Fill any empty place with $0$ , and write out as many digits of a repeating block as you need. Then read from left to right. At the first place where the digits differ, the number with the greater digit is greater. For example, $0.45 < 0.5$ , because $0.45$ and $0.50$ first differ in the tenths place, where $4 < 5$ .

This check works for decimals in the forms that division gives (terminating, or repeating with a block that is not just the digit $9$ ). In those forms, one unit of any place is worth more than all the digits after that place put together, so the later digits cannot make up the difference.

For two negative decimals, compare their distances from $0$ in the same way. For example, the distance of $-0.67$ from $0$ is $0.67$ . The number that is farther from $0$ is the lesser one.

Worked example

Comparing a negative fraction and a negative decimal

Compare $-\frac{2}{3}$ and $-0.67$ . Which number is greater?

Check the signs first. Both numbers are negative, so the signs alone cannot decide. One number is a fraction and the other is a decimal, so write the fraction as a decimal. Divide 2 by 3. After the decimal point, every step is 20 divided by 3, which is 6 with remainder 2, so the digit 6 repeats forever.
$\frac{2}{3} = 0.\overline{6}, \quad \text{so} \quad -\frac{2}{3} = -0.\overline{6}$
Keep the exact repeating form. Rounded to the nearest hundredth, two thirds is about 0.67. That rounded value would make the two numbers look equal when they are not.
$\frac{2}{3} \approx 0.67$
Compare the distances from 0 place by place. Write 0.67 as 0.670 and write out three digits of the repeating decimal. The ones digits match and the tenths digits match. In the hundredths place, 6 is less than 7.
$0.666\ldots < 0.670, \quad \text{so} \quad 0.\overline{6} < 0.67$
Both numbers are negative, so the one farther from 0 is the lesser one. Negative 0.67 is farther from 0, so it lies to the left of negative two thirds.
$-0.67 < -0.\overline{6}, \quad \text{so} \quad -0.67 < -\frac{2}{3}$

Answer

$-\frac{2}{3}$ is greater than $-0.67$ , so $-\frac{2}{3} > -0.67$ .

Key concept

Same form, then compare

The order of rational numbers is their order on the number line: farther left means less. To compare two rational numbers:

Check the signs. Any negative number is less than $0$ , and $0$ is less than any positive number.
If the signs match, use the same form. Write both numbers as fractions with the same positive denominator $d$ and integer numerators $a$ and $b$ . Then the numerators decide, as the formula shows. Or write both numbers as exact decimals and compare them place by place.
Take care with negatives. For two negative numbers, the one farther from $0$ is the lesser one.

\frac{a}{d} < \frac{b}{d} \ \text{ exactly when } \ a < b \qquad (d > 0)

Worked example

Ordering a mixed list

Order these numbers from least to greatest: $-1.3$ , $\frac{3}{5}$ , $-\frac{4}{3}$ , $0.\overline{5}$ , $-\frac{5}{4}$ .

Sort by sign. Three numbers are negative and two are positive. Every negative number is less than every positive number, so the three negative numbers come first.
$\text{negative: } -1.3,\ -\frac{4}{3},\ -\frac{5}{4} \qquad \text{positive: } \frac{3}{5},\ 0.\overline{5}$
Write each fraction as an exact decimal by dividing its numerator by its denominator. Dividing 3 by 5 gives exactly 0.6, and dividing 5 by 4 gives exactly 1.25. Dividing 4 by 3 gives 1.333..., where the digit 3 repeats forever because every step leaves remainder 1. Keep each negative sign.
$\frac{3}{5} = 0.6, \qquad -\frac{4}{3} = -1.\overline{3}, \qquad -\frac{5}{4} = -1.25$
Compare the distances of the three negative numbers from 0. Write them as 1.300, 1.333... and 1.250. All three have 1 in the ones place. In the tenths place, 2 is less than 3, so 1.25 is the smallest distance. Then 1.300 and 1.333... first differ in the hundredths place, where 0 is less than 3.
$1.25 < 1.3 < 1.\overline{3}$
For negative numbers, the one farther from 0 is the lesser one. So the order of the negative numbers is the reverse of the order of their distances.
$-1.\overline{3} < -1.3 < -1.25$
Compare the two positive numbers. Write the repeating decimal as 0.555... and write 0.6 as 0.600. They first differ in the tenths place, where 5 is less than 6.
$0.\overline{5} < 0.6$
Join the two groups with the negative numbers first. Write each number in its original form.
$-\frac{4}{3} < -1.3 < -\frac{5}{4} < 0.\overline{5} < \frac{3}{5}$

Answer

From least to greatest, the order is $-\frac{4}{3}$ , $-1.3$ , $-\frac{5}{4}$ , $0.\overline{5}$ , $\frac{3}{5}$ .

Takeaway. To compare or order rational numbers, think about where they sit on the number line. Check the signs first. Then write numbers that have the same sign in the same form: fractions with a common positive denominator, or exact decimals with a bar over any repeating block. For negative numbers, farther from $0$ means less. Next, you will use these same forms to add and subtract rational numbers.